Here I think the teacher made a huge mistake, calling you both out in front of the class. They know how kids can be and an event like this can trigger bullying easily. This case should have been dealt with privately.
As about public and private school, in my country almost everyone goes to public school. Most likely there are private schools as well, but in 99% of the cases, that is out of the question, there's no need for that.
greetings, I honestly do not believe that this was the teacher's purpose, but I still hope that today she handles this type of situation better, because what generated this situation was sad. I believe that we will live our experiences and life lessons no matter where we are. that's what it's all about. ❤️